Time Series Analysis

PART II

• Econometric Forecasting

• Forecasting is an important part of econometric analysis, for some people probably the most important. How do we forecast economic variables, such as GDP, inflation, exchange rates, stock prices, unemployment rates, and myriad other economic variables?

APPROACHES TO ECONOMIC FORECASTING

(1)exponential smoothing methods

(2) single-equation regression models

(3) simultaneous-equation regression models

(4) Autoregressive integrated moving average models (ARIMA)

(5) vector autoregression.

• ARIMA Models

• Popularly known as the Box–Jenkins (BJ) methodology, but technically known as the

ARIMA methodology, the emphasis of these methods is not on constructing single-equation or simultaneous-equation models but on analyzing the probabilistic, or stochastic, properties of economic time series on their own under the philosophy let the data speak for themselves.

Unlike the regression models, in which Yt is explained by k regressor X1, X2, X3, . . . , Xk, the BJ-type time series models allow Yt to be explained by past, or lagged, values of Y itself and stochastic error terms.

Autoregressive processes (AR)

An AR(1) process is written as:

yt = yt-1 + t where t ~ IID(0,2)

ie. the current value of yt is equal to times its previous value plus an unpredictable component t . we say that Yt follows a first-order autoregressive, or AR(1), stochastic process.

Here the value of Y at time t depends on its value in the previous time period and a random term; the Y values are expressed as deviations from their mean value. In other words, this model says that the forecast value of Y at time t is simply some proportion (= ) of its value at time (t − 1) plus a random shock or disturbance at time t.

If we consider a model

yt = 1yt-1 + 2yt-2 + t

then we say that Yt follows a second-order autoregressive, or AR(2), process. That is, the value of Y at time t depends on its value in the previous two time periods.

This can be extended to an AR(p) process

yt = 1yt-1 + 2yt-2 +…..+pyt-p + t

where t ~ IID(0,2)

ie. the current value of yt depends on p past values plus an unpredictable component t .

Moving Average processes(MA)

An MA(1) process is written as

  yt = t + β1t-1 where t ~ IID(0,2)

ie. the current value is given by an unpredictable component t and β times the previous period’s error.

• Here Y at time t is equal to a constant plus a moving average of the current and past error terms. Thus, in the present case, we say that Y follows a first-order moving average, or an MA(1), process.

But if Y follows the expression,

yt = t + β1t-1 + β2t-2

then it is an MA(2) process, This can also be extended to an MA(q) process

yt = t + β 1t-1 +…..+ β qt-q

where t ~ IID(0,2) In short, a moving average process is simply a

linear combination of white noise error terms.

Autoregressive and Moving Average (ARMA) Process

Of course, it is quite likely that Y has characteristics of both AR and MA and is therefore ARMA. Thus, Yt follows an ARMA(1, 1) process if it can be written as

yt = 1yt-1 + t + β1t-1

Because there is one autoregressive and one moving average term.

In general, in an ARMA( p, q) process, there will be p autoregressive and q moving average terms,

yt = 1yt-1 +...+ pyt-p +t + β1t-1 +...+ βqt-q

• Autoregressive Integrated Moving Average (ARIMA) Process

• The time series models we have already discussed are based on the assumption that the time series involved are (weakly) stationary . Briefly, the mean and variance for a weakly stationary time series are constant and its covariance is time-invariant. But we know that many economic time series are nonstationary, that is, they are integrated.

if a time series is integrated of order 1 [i.e., it is I(1)], its first differences are I(0), that is, stationary. Similarly, if a time series is I(2), its second difference is I(0). In general, if a time series is I(d), after differencing it d times we obtain an I(0) series. Therefore, if we have to difference a time series d times to make it stationary and then apply the ARMA(p, q) model to it,

we say that the original time series is ARIMA(p, d, q), that is, it is an autoregressive integrated moving average time series, where p denotes the number of autoregressive terms, d the number of times the series has to be differenced before it becomes stationary, and q the number of moving average terms.

Thus, an ARIMA(2, 1, 2) time series has to be differenced once (d = 1) before it becomes stationary and the (first-differenced) stationary time series can be modeled as an ARMA(2, 2) process, that is, it has two AR and two MA terms.

• THE BOX–JENKINS (BJ) METHODOLOGY• How does one know whether it follows a purely

AR process (and if so, what is the value of p) or a purely MA process (and if so, what is the value of q) or an ARMA process (and if so, what are

the values of p and q) or an ARIMA process, in which case we must know the values of p, d, and q. The BJ methodology comes in handy in answering the preceding question. The method consists of four steps:

• Step 1. Identification.

To find out the appropriate values of p, d,

and q. We use the correlogram and partial correlogram for this task.

• Step 2. Estimation. Having identified the appropriate p and q

values, the next stage is to estimate the parameters of the autoregressive and moving

average terms included in the model. Sometimes this calculation can be done by simple least squares but sometimes we will have to resort to nonlinear (in parameter) estimation methods.

• Step 3. Diagnostic checking.• Having chosen a particular ARIMA model,

and having estimated its parameters, we next see whether the chosen model fits the data reasonably well, for it is possible that another ARIMA model might do the job as well. This is why Box–Jenkins ARIMA modeling is more an art than a science; considerable skill is required to choose the right ARIMA model.

• One simple test of the chosen model is to see if the residuals estimated from this model are white noise; if they are, we can accept the particular fit; if not, we must start over. Thus, the BJ methodology is an iterative process.

• Step 4. Forecasting.

• One of the reasons for the popularity of the ARIMA modeling is its success in forecasting. In many cases, the forecasts obtained by this method are more reliable than those obtained from the traditional econometric modeling, particularly for short-term forecasts. Of course,

each case must be checked.

Vector ARX

The term autoregressive is due to the appearance of the lagged value of the

dependent variable on the right-hand side and the term vector is due to the

fact that we are dealing with a vector of two (or more) variables.

Volatility

Financial time series, such as stock prices, exchange rates, inflation rates, etc. often exhibit the phenomenon of volatility clustering, that is, periods in which their prices show wide swings for an extended time period followed by periods in which there is

relative calm

Knowledge of volatility is of crucial importance in many areas. For example, considerable macroeconometric work has been done in studying the variability of inflation over time. For some decision makers, inflation in itself may not be bad, but its variability is bad because it makes financial planning difficult.

• A characteristic of most of thefinancial time series is that in their level form they are random walks; that is, they are nonstationary.

On the other hand, in the first difference form, they are generally stationary

Therefore, instead of modeling the levels of financial time series, why not model their first differences? But these first differences often exhibit wide swings, or volatility, suggesting that the variance of financial time series varies over time.

How can we model such “varying variance”? This is where the so-called autoregressive conditional heteroscedasticity (ARCH)